Remove and discard the tough outer layers of the leeks. Cut the leeks open and in half lengthwise. Wash very well to remove any sand; dry with paper towels. Cut into 1-inch pieces across the leek stalks.

Soak the mushrooms in hot water for 20 minutes, or until soft; drain. Rinse in warm water; drain. Squeeze out any excess moisture. Remove and discard the stems.

Heat a large stockpot and add the vegetable oil. Add the leeks and stir-fry for 1 minute. Add 4 cups water, bring to a boil, and add the black mushrooms, soybean sprouts, carrot, salt, and pepper. Return to boiling again, turn the heat down to medium-low, and cook, covered, for 1 hour. Strain. Cool and store in the refrigerator for up to 5 days or freeze in a tightly sealed container for up to 3 months.